The Bennequin bound [1] says that, for a transverse knot (or later link) $K$ in $S^3$, $$\mathrm{sl}(K) \le - \chi(\Sigma)$$ for any Seifert surface $\Sigma$ for $K$, where $\mathrm{sl}$ is the self-linking number.

Lisca and Matic proved [2, Theorem 3.4] that for a Legendrian knot $K$ $$|r(K)| + \mathrm{tb}(K) \le 2g(\Sigma) - 1$$ for any smooth surface $\Sigma \subset B^4$ with boundary $K$, where $\mathrm{tb}$ and $r$ are the Thurston-Bennequin and rotation numbers.

EDIT: This was actually proved by Rudolph [3].

I have several questions here.

(a) Why is the Lisca-Matic bound always stated in terms of Legendrian knots? The combination $|r| + \mathrm{tb}$ is exactly the self-linking number of the transverse push-off, with one of the two orientations. Wouldn't it be simpler to just state it in terms of transverse knots?

(b) Isn't the Lisca-Matic bound strictly stronger than the Bennequin bound? Why isn't it more popular to quote it? Maybe it's about the extension to links (which, in the case of the Bennequin bound, is due to Eliashberg)? It seems that Lisca and Matic only state the result for knots in their paper, but it seems to me that the proof should carry over.

For knots/links in more general 3-manifolds, the Bennequin bound applies as long as you have a tight contact structure, while to state Lisca-Matic you need some sort of filling.

[1] Bennequin, "Entrelacements et équations de Pfaff", In Third Schnepfenried Geometry Conference, Vol. 1, Schnepfenried, 1982, 87–161. Astérisque 107. Paris: Société Mathématique de France, 1983.

[2] Lisca, P., and G. Matić. “Stein 4-manifolds with Boundary and Contact Structures.” 55–66. Topology and Its Applications 88, nos. 1–2, 1998.

[3] Rudolph, Lee, "Quasipositivity as an obstruction to sliceness." Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51–59.